MinT - Best Known (139−95, 139, s)-Nets in Base 5

Best Known (139−95, 139, s)-Nets in Base 5

(139−95, 139, 78)-Net over F₅ — Constructive and digital

Digital (44, 139, 78)-net over F₅, using

t-expansion [i] based on digital (38, 139, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(139−95, 139, 84)-Net over F₅ — Digital

Digital (44, 139, 84)-net over F₅, using

t-expansion [i] based on digital (43, 139, 84)-net over F₅, using
- net from sequence [i] based on digital (43, 83)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 43 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(139−95, 139, 484)-Net in Base 5 — Upper bound on s

There is no (44, 139, 485)-net in base 5, because

1 times m-reduction [i] would yield (44, 138, 485)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3 125978 390651 168171 870712 460925 577238 269724 056261 015089 503734 942714 783322 668155 113014 427859 284445 > 5¹³⁸ [i]